An Ostrowski Type Inequality for Mappings Whose Second Derivatives are Bounded and Applications

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An Ostrowski Type Inequality for Mappings Whose Second Derivatives are Bounded and Applications

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Dragomir, Sever S and Barnett, Neil S (1998) An Ostrowski Type Inequality for Mappings Whose Second Derivatives are Bounded and Applications. RGMIA research report collection, 1 (2).

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RGMIA Research Report Collection, Vol. 1, No. 2, 1998 http://sci.vut.edu.au/∼rgmia

AN OSTROWSKI TYPE INEQUALITY FOR MAPPINGS WHOSE SECOND DERIVATIVES ARE BOUNDED AND APPLICATIONS

S.S. DRAGOMIR AND N.S. BARNETT

Abstract. An integral inequality of Ostrowski’s type for mappings whose second derivatives are bounded is proved. Applications in Numerical Integration and for special means are pointed out.

1 Introduction

In [1],S.S. Dragomir and S. Wang obtained the following Ostrowski type inequality [2, p. 468]:

Theorem 1.1. Let f : [a, b]→R be continuous on[a, b] and a differentiable on(a, b). If f⁰ ∈ L1(a, b)and there exists the constants γ,Γ so that

γ≤f⁰(x)≤Γ for all x∈(a, b), (1.1)

then we have the inequality:

f(x)− 1 b−a

Zb a

f(t)dt−f(b)−f(a) b−a

x−a+b 2

≤ 1

4(b−a) (Γ−γ) (1.2)

for allx∈[a, b].

The proof used essentially the identity f(x) = 1

b−a Zb a

f(t)dt+ 1 b−a

Zb a

p(x, t)f⁰(t)dt (1.3)

for allx∈[a, b],wheref is as above and the kernelp(·,·) : [a, b]²→Ris given by p(x, t) :=





t−a ift∈[a, x]

t−b ift∈(x, b]

(1.4)

and Gr¨uss’ integral inequality which says (see for example [1]) that:

1 b−a

Zb a

g(x)h(x)dx− 1 b−a

Zb a

g(x)dx· 1 b−a

Zb a

h(x)dx (1.5)

≤ 1

4(Φ−ϕ) (Γ−γ) provided g, h: [a, b]→Rare integrable and

ϕ≤g(x)≤Φ, γ≤h(x)≤Γ (1.6)

for allx∈[a, b].

The main aim of this paper is to point out a new estimation of the left membership of (1.2) and to apply it for special means and in Numerical Integration.

Date.December, 1998

1991 Mathematics Subject Classification. Primary 26 D 15; Secondary 41 A 55.

Key words and phrases.Ostrowski’s Inequality, Numerical Integration, Special Means.

67

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2 A New Integral Inequality The following results holds:

Theorem 2.1. Let f : [a, b] → R be continuous on [a, b] and twice differentiable on (a, b), whose second derivativef⁰⁰: (a, b)→R is bounded on(a, b). Then we have the inequality

f(x)− 1 b−a

Zb a

f(t)dt−f(b)−f(a) b−a

x−a+b 2

(2.1)

≤ 1 2





"

x−^a+b2

² (b−a)² +1

4

#² + 1

12



(b−a)²kf⁰⁰k_∞

≤ kf⁰⁰k_∞

6 (b−a)² for allx∈[a, b].

Proof. For the sake of completness, we give a short proof of the identity (1.3) which will be used in the sequel.

Integrating by parts, we have Zx a

(t−a)f⁰(t)dt= (x−a)f(x)− Zx a

f(t)dt and

Zb x

(t−b)f⁰(t)dt= (b−x)f(x)− Zb x

f(t)dt.

Adding these two equalities, we get Zx

a

(t−a)f⁰(t)dt+ Zb x

(t−b)f⁰(t)dt= (b−a)f(x)− Zb a

f(t)dt which is equivalent to (1.3).

Applying the identity (1.3) forf⁰(·) we can state f⁰(t) = 1

b−a Zb a

f⁰(s)ds+ 1 b−a

Zb a

p(t, s)f⁰⁰(s)ds which is equivalent to

f⁰(t) = f(b)−f(a)

b−a + 1

b−a Zb a

p(t, s)f⁰⁰(s)ds.

Substitutingf⁰(t) in the right membership of (1.3) we get f(x) = 1

b−a Zb a

f(t)dt

+ 1 b−a

Zb a

p(x, t)



f(b)−f(a)

b−a + 1

b−a Zb a

p(t, s)f⁰⁰(s)ds



dt

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Ostrowski for Bounded Second Derivatives Mappings 69

= 1

b−a Zb a

f(t)dt+f(b)−f(a) (b−a)²

Zb a

p(x, t)dt

+ 1

(b−a)² Zb a

Zb a

p(x, t)p(t, s)f⁰⁰(s)dsdt and as

Zb a

p(x, t)dt= Zx a

(t−a)dt+ Zb x

(t−b)dt

= (b−a)

x−a+b 2

we get the integral identity:

f(x) = 1 b−a

Zb a

f(t)dt+f(b)−f(a) b−a

x−a+b 2

(2.2)

+ 1

(b−a)² Zb a

Zb a

p(x, t)p(t, s)f⁰⁰(s)dsdt

Now, using the identity (2.2) we get

f(x)− 1 b−a

Zb a

f(t)dt− f(b)−f(a) b−a

x−a+b 2

(2.3)

≤ 1 (b−a)²

Zb a

|p(x, t)p(t, s)| |f⁰⁰(s)|dsdt

≤ kf⁰⁰k_∞ (b−a)²

Zb a

|p(x, t)| |p(t, s)|dsdt.

We have

Zb a

|p(t, s)|ds= (t−a)²+ (b−t)²

2 .

Also

A:=

Zb a

|p(x, t)|

"

(t−a)²+ (b−t)² 2

# dt

=1 2



 Zx a

(t−a) h

(t−a)²+ (b−t)² i

dt+ Zb x

(b−t) h

(t−a)²+ (b−t)² i

dt





RGMIA Research Report Collection, Vol. 1, No. 2, 1998

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= 1 2



 Zx a

h

(t−a)³+ (t−a) (b−t)²i dt+

Zb x

h

(t−a)²(b−t) + (b−t)³i dt



.

Note that

Zx a

(t−a)³dt= (x−a)⁴

4 ,

Zx a

(t−a) (b−t)²dt

=−1

3(b−x)³(x−a)− 1

12(b−x)⁴+ 1

12(b−x)⁴; Zb

x

(t−b) (t−a)²dt

=1

3(x−a)³(b−x)− 1

12(b−a)⁴+ 1

12(x−a)⁴; Zb

x

(t−b)³dt=(x−b)⁴

4 .

Consequently, we have A=1

2

"

(x−a)⁴

4 −1

3(b−x)³(x−a)− 1

12(b−x)⁴+ 1

12(b−a)⁴

−1

3(x−a)³(b−x) + 1

12(b−a)⁴− 1

12(x−a)⁴+(x−b)⁴ 4

#

= 1 12

h

(x−a)⁴−2 (b−x)³(x−a)−2 (x−a)³(b−x)

+ (b−x)⁴+ (b−a)⁴ i

. Now, observe that,

(x−a)⁴+ (b−x)⁴=h

(x−a)²+ (b−x)²i2

−2 (x−a)²(b−x)² and

−2 (b−x)³(x−a)−2 (x−a)³(b−x)

=−2 (x−a) (b−x)h

(x−a)²+ (b−x)²i

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then

B := 12A= h

(x−a)²+ (b−x)² i2

−2 (x−a) (b−x) h

(x−a)²+ (b−x)² i

−2 (x−a)²(b−x)²+ (b−a)⁴

= h

(x−a)²+ (b−x)²−(x−a) (b−x) i2

−3 (x−a)²(b−x)²+ (b−a)⁴. But a simple calculation shows that

(x−a)²+ (b−x)²= 1

2(b−a)²+ 2

x−a+b 2

2

and as

(x−a)²+ (b−x)²+ 2 (x−a) (b−x) = (b−a)² we get

2 (x−a) (b−x) = (b−a)²− h

(x−a)²+ (b−x)² i

i.e.,

(x−a) (b−x) =1

2(b−a)²−1 2 h

(x−a)²+ (b−x)²i

=1

4(b−a)²−

x−a+b 2

2

. Consequently,

B=

"

1

2(b−a)²+ 2

x−a+b 2

2

−1

4(b−a)²+

x−a+b 2

2#2

−

−3

"

1

4(b−a)²−

x−a+b 2

2#2

+ (b−a)⁴

= 6

x−a+b 2

2

+ 3 (b−a)²

x−a+b 2

2

+7

8(b−a)⁴ and then

A= 1 12

"

6

x−a+b 2

4

+ 3 (b−a)²

x−a+b 2

2

+7

8(b−a)⁴

# .

Now, using the inequality (2.3) and simple algebraic manipulations, we get the first result in (2.1).

The second part is obvious by the fact that

x−a+b 2

≤b−a 2 for allx∈[a, b].

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3 Applications in Numerical Integration

LetIh :a=x0< x1 < ... < xn−1 < xn =b be a division of the interval [a, b], ξ_i ∈[xi, xi+1] (i= 0,1, ..., n−1) a sequence of intermediate points andh_i:=x_i+1−x_i(i= 0,1, ..., n−1). As in [1],consider the perturbed Riemann’s sum defined by

AG(f, Ih,ξ) :=

nX−1 i=0

f(ξ_i)hi−

nX−1 i=0

ξ_i−x_i+x_i+1 2

f(xi+1)−f(xi) . (3.1)

In that paper Dragomir and Wang proved the following result:

Theorem 3.1. Let f : [a, b] → R be continuous on [a, b] and differentiable on (a, b), whose derivative f⁰ : (a, b)→Ris bounded on (a, b)and assume that

γ≤f⁰(x)≤Γ for all x∈(a, b). (3.2)

Then we have the quadrature formula:

Zb a

f(x)dx=AG(f, Ih,ξ) +RG(f, Ih,ξ) (3.3)

and the remainderRG(f, Ih,ξ)satisfies the estimation

|RG(f, Ih,ξ)| ≤ 1

4(Γ−γ)

nX−1 i=0

h²_i, (3.4)

for allξ= ξ₀, ..., ξ_n₋₁

as above.

Here, we prove another type of estimation for the remainder RG(f, Ih,ξ) in the case when f is twice differentiable.

Theorem 3.2. Let f : [a, b] → R be continuous on [a, b] and twice differentiable on (a, b), whose second derivativef⁰⁰: (a, b)→Ris bounded on(a, b). Denotekf⁰⁰k_∞:= sup

t∈(a,b)|f⁰⁰(t)|<

∞. Then we have the quadrature formula (3.3) and the remainder RG(f, Ih,ξ) satisfies the estimation:

|RG(f, Ih,ξ)| ≤ kf⁰⁰k_∞ 2

nX−1 i=0













ξ_i−^xⁱ^+x2ⁱ⁺¹

2

h²_i +1 4





2

+ 1 12







 h³_i (3.5)

≤ kf⁰⁰k_∞ 6

nX−1 i=0

h³_i for allξ_i as above.

Proof. Apply Theorem 2.1 on the interval [xi, xi+1] (i= 0, . . . , n−1) to obtain

f(ξ_i)hi−

xZ_i+1

xi

f(t)dt−

ξ_i−xi+xi+1

2

(f(xi+1)−f(xi))

≤ kf⁰⁰k_∞ 2











ξ_i−^xⁱ^+x₂ⁱ⁺¹2

h²_i +1 4





2

+ 1 12





h³_i ≤kf⁰⁰k_∞ 6 h³_i for all ξ_i∈[xi, xi+1] and i∈ {0, . . . , n−1}.

Summing over i from 0 ton−1 and using the generalized triangle inequality, we get the desired inequality (3.5).

We omit the details.

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4 Applications for Special Means Recall the following means :

(a) The arithmetic mean

A=A(a, b) := a+b

2 , a, b≥0;

(b) The geometric mean:

G=G(a, b) :=√

ab, a, b≥0;

(c) The harmonic mean:

H =H(a, b) := 2 1 a+1

b

, a, b≥0;

(d) The logarithmic mean:

L=L(a, b) :=







a ifa=b

b−a

lnb−lna ifa6=b

a, b >0;

(e) The identric mean:

I:=I(a, b) =









a ifa=b

1 e

b^b a^a

_b−a¹

ifa6=b

a, b >0;

(f) Thep-logarithmic mean:

Lp =Lp(a, b) :=









b^p+1−a^p+1 (p+ 1) (b−a)

¹_p

ifa6=b;

a ifa=b

where p∈R\ {−1,0}anda, b >0.

The following simple relationships are well known in the literature H ≤G≤L≤I≤A

(4.1) and

L_p is monotonically increasing inp∈RwithL₀:=I andL₋₁:=L.

(4.2)

1. Consider the mappingf(x) =x^p (p≥2) on [a, b]⊂(0,∞). Applying the inequality (2.1) forf(x) =x^p,we get:

x^p−L^p_p−pL^p_p⁻₋¹₁(x−A) (4.3)

≤ p(p−1)b 2

p−2((x−A)² (b−a)² +1

4 2

+ 1 12

)

(b−a)²

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≤ p(p−1)b^p⁻²

6 (b−a)² for allx∈[a, b].

Choosing in(4.3), x=A,we get

0≤L^p_p−A^p≤ 7

96p(p−1)b^p⁻²(b−a)². (4.4)

2. Consider the mappingf(x) =_x¹ (x∈[a, b]⊂(0,∞)). Applying the inequality (2.1) for this mapping we get:

1 x− 1

L−x−A G²

(4.5)

≤ 1 3a³

((x−A)² (b−a)² +1

4 ²

+ 1 12

)

(b−a)²≤ 1

3a³(b−a)² for allx∈[a, b].

Choosing in (4.5) x=A, we get

0≤A−L

AL ≤ 7

48a³(b−a)². (4.6)

Also, choosing in (4.5)x=L,we get

0≤ A−L G² ≤ 1

3a³

((x−A)² (b−a)² +1

4 ²

+ 1 12

)

(b−a)² (4.7)

≤ 1

3a³(b−a)².

3. Finally, let us consider the mapping f(x) =−lnx(x∈[a, b]⊂(0,∞)). Then, by (2.1), we get:

ln





 I

b a

^x−A_b−a x







≤ 1 2a²

((x−A)² (b−a)² +1

4 ²

+ 1 12

)

(b−a)²≤ 1

6a²(b−a)² (4.8)

Puttingx=Ain (4.8) we get 1≤ A

I ≤exp 7

96a²(b−a)²

. (4.9)

Acknowledgement. The authors would like to thank Professor Peter Cerone for his suggestions in improving this paper.

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References

[1] S.S. DRAGOMIR, and S. WANG, An inequality of Ostrowski-Gr¨uss’ type and its applications to the estimation of error bounds for some special means and for some numerical quadrature rules,Computers Math. Applic.33(1997), 15-20.

[2] D.S. MITRINOVI ´C, J.E. PE ˘CARI ´C and A.M. FINK,Inequalities for Functions and Their Integrals and Derivatives, Kluwer Academic Publisher, Dordrecht, 1994.

School of Communications and Informatics, Victoria University of Technology, PO Box 14428, Melbourne City MC, Victoria 8001, Australia.

E-mail address: {sever, neil}@matilda.vut.edu.au

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